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'''Algebra''' (from [[Arabic Language|Arabic]]: الجبر‎, transliterated "al-jabr", meaning "reunion of broken parts") is a part of [[mathematics]] (often called '''math''' in the United States and '''maths''' or numeracy in the [[United Kingdom]]<ref>{{cite web|title=Math or Maths|url=http://mathworld.wolfram.com/Mathematics.html|publisher=Wolfram.com|accessdate=11 April 2013}}</ref> ). It uses [[variable]]s to represent a value that is not yet known.  When an equals sign (=) is used, this is called an [[equation]].  A very simple equation using a variable is: <tt>2 + 3 = x.</tt> In this example, <tt>x = 5</tt>, or it could also be said that "<code>x</code> equals five".  This is called ''solving for <code>x</code>''.<ref>{{cite web|title=Algebra Introduction|url=http://www.mathsisfun.com/algebra/introduction.html|publisher=Math Is Fun|accessdate=11 April 2013}}</ref>
{{Short description|Area of mathematics}}
{{about||the kind of algebraic structure|Algebra over a field|other uses}}
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[[File:Quadratic formula.svg|thumb|The [[quadratic formula]] expresses the solution of the equation {{math|1=''ax''<sup>2</sup> + ''bx'' + ''c'' = 0}}, where {{mvar|a}} is not zero, in terms of its coefficients {{math|''a'', ''b''}} and {{mvar|c}}.]]


Besides equations, there are [[inequality|inequalities]] (''less than'' and ''greater than''). A special type of equation is called the function. This is often used in making [[graph]]s because it always turns one input into one output.
'''Algebra''' ({{etymology|ar|''{{wikt-lang|ar|جبر|الجبر}}'' ({{transl|ar|al-jabr}})|reunion of broken parts,<ref name=oed>{{cite Lexico|algebra|access-date=2013-11-20}} {{Cite web |url=http://www.oxforddictionaries.com/us/definition/english/algebra |title=Archived copy |access-date=2013-11-20 |archive-date=2013-12-31 |archive-url=https://web.archive.org/web/20131231173558/http://www.oxforddictionaries.com/us/definition/english/algebra |url-status=bot: unknown }}.</ref> [[bonesetter|bonesetting]]}})<ref name="CRC Press">{{cite book |url=https://books.google.com/books?id=3mlQDwAAQBAJ&q=bonesetting+algebra&pg=PA722 |title=Abstract Algebra: A Comprehensive Treatment |last1=Menini|first1=Claudia |last2=Oystaeyen|first2=Freddy Van |date=2017-11-22 |publisher=[[CRC Press]] |isbn=978-1-4822-5817-2 |language=en |access-date=2020-10-15 |archive-date=2021-02-21 |archive-url=https://web.archive.org/web/20210221075950/https://books.google.com/books?id=3mlQDwAAQBAJ&q=bonesetting+algebra&pg=PA722 |url-status=live}}</ref> is one of the [[areas of mathematics|broad areas]] of [[mathematics]]. Roughly speaking, algebra is the study of [[mathematical symbol]]s and the rules for manipulating these symbols in [[formula]]s;<ref>See {{harvnb|Herstein|1964}}, page 1: "An algebraic system can be described as a set of objects together with some operations for combining them".</ref> it is a unifying thread of almost all of mathematics.<ref>See {{harvnb|Herstein|1964}}, page 1: "...it also serves as the unifying thread which interlaces almost all of mathematics".</ref>


Algebra can be used to solve real problems because the rules of algebra work in real life and numbers can be used to represent the values of real things. [[Physics]], [[engineering]] and [[computer programming]] are areas that use algebra all the time.  It is also useful to know in [[surveying]], [[construction]] and [[business]], especially [[accounting]].  
[[Elementary algebra]] deals with the manipulation of [[variable (mathematics)|variables]] as if they were numbers (see the image), and is therefore essential in all applications of mathematics. [[Abstract algebra]] is the name given in [[education]] to the study of [[algebraic structure]]s such as [[group (mathematics)|groups]], [[ring (mathematics)|rings]], and [[field (mathematics)|fields]]. [[Linear algebra]], which deals with [[linear equation]]s and [[linear mapping]]s, is used for modern presentations of [[geometry]],<!-- Berger's ''Geometry'' must be cited here  --> and has many practical applications (in [[weather forecasting]], for example). There are many areas of mathematics that belong to algebra, some having "algebra" in their name, such as [[commutative algebra]] and some not, such as [[Galois theory]].  


People who do algebra use the rules of [[number]]s and mathematic [[operation (mathematics)|operations]] used on numbers. The simplest are [[addition|adding]], [[subtraction|subtracting]], [[multiplication|multiplying]], and [[division (mathematics)|dividing]]. More advanced operations involve [[exponent]]s, starting with [[square (algebra)|squares]] and [[square root]]s.
The word ''algebra'' is not only used for naming an area of mathematics and some subareas; it is also used for naming some sorts of algebraic structures, such as an [[algebra over a field]], commonly called an ''algebra''. Sometimes, the same phrase is used for a subarea and its main algebraic structures; for example, [[Boolean algebra]] and a [[Boolean algebra (structure)|Boolean algebra]]. A mathematician specialized in algebra is called an algebraist.


Algebra was first used to solve [[algebraic equation|equations]] and inequalities. Two examples are [[linear equation]]s (the equation of a straight line, <tt>y=mx+b</tt> <big>or</big> <code>y=mx+c</code>) and [[quadratic equation]]s, which has variables that are [[Square (algebra)|squared]] (multiplied by itself, for example: <tt>2*2, 3*3, or x*x</tt>).
== Etymology ==
[[File:Muḥammad ibn Mūsā al-Khwārizmī.png|thumb|upright=0.8|The word ''algebra'' comes from the title of a book by [[Muhammad ibn Musa al-Khwarizmi]].<ref>Esposito, John L. (2000-04-06). ''The Oxford History of Islam''. Oxford University Press. p. 188. {{ISBN|978-0-19-988041-6}}.</ref>]]
 
The word ''algebra'' comes from the {{lang-ar|الجبر|lit=reunion of broken parts,<ref name="oed" /> [[bonesetting]]<ref name="CRC Press" />|translit=al-jabr}} from the title of the early 9th century book ''[[The Compendious Book on Calculation by Completion and Balancing|<sup>c</sup>Ilm al-jabr wa l-muqābala]]'' "The Science of Restoring and Balancing" by the [[Persian people|Persian]] mathematician and astronomer [[Muḥammad ibn Mūsā al-Khwārizmī|al-Khwarizmi]]. In his work, the term ''al-jabr'' referred to the operation of moving a term from one side of an equation to the other, المقابلة ''al-muqābala'' "balancing" referred to adding equal terms to both sides. Shortened to just ''algeber'' or ''algebra'' in Latin, the word eventually entered the English language during the 15th century, from either Spanish, Italian, or [[Medieval Latin]]. It originally referred to the surgical procedure of setting [[Broken bone|broken]] or [[Dislocated|dislocated bones]]. The mathematical meaning was first recorded (in English) in the 16th century.<ref>{{cite encyclopedia|title=Algebra|editor=T. F. Hoad|encyclopedia=The Concise Oxford Dictionary of English Etymology|publisher=Oxford University Press|location=Oxford|year=2003|url=https://archive.org/details/conciseoxforddic00tfho|url-access=subscription|doi=10.1093/acref/9780192830982.001.0001|isbn=978-0-19-283098-2}}</ref>
 
== Different meanings of "algebra" ==
The word "algebra" has several related meanings in mathematics, as a single word or with qualifiers.
* As a single word without an article, "algebra" names a broad part of mathematics.
* As a single word with an article or in the plural, "an algebra" or "algebras" denotes a specific mathematical structure, whose precise definition depends on the context. Usually, the structure has an addition, multiplication, and scalar multiplication (see [[Algebra over a field]]). When some authors use the term "algebra", they make a subset of the following additional assumptions: [[Associative property|associative]], [[Commutative property|commutative]], [[Unital algebra|unital]], and/or finite-dimensional. In [[universal algebra]], the word "algebra" refers to a generalization of the above concept, which allows for [[Operation (mathematics)|n-ary operations]].
* With a qualifier, there is the same distinction:
** Without an article, it means a part of algebra, such as [[linear algebra]], [[elementary algebra]] (the symbol-manipulation rules taught in elementary courses of mathematics as part of [[primary education|primary]] and [[secondary education]]), or [[abstract algebra]] (the study of the algebraic structures for themselves).
** With an article, it means an instance of some algebraic structure, like a [[Lie algebra]], an [[associative algebra]], or a [[vertex operator algebra]].
** Sometimes both meanings exist for the same qualifier, as in the sentence: ''[[Commutative algebra]] is the study of [[commutative ring]]s, which are [[algebra (ring theory)|commutative algebras]] over the integers''.
 
== Algebra as a branch of mathematics ==
 
Algebra began with computations similar to those of [[arithmetic]], with letters standing for numbers.<ref name=citeboyer /> This allowed proofs of properties that are true no matter which numbers are involved. For example, in the [[quadratic equation]]
:<math>ax^2+bx+c=0,</math>
<math>a, b, c</math> can be any numbers whatsoever (except that <math>a</math> cannot be <math>0</math>), and the [[quadratic formula]] can be used to quickly and easily find the values of the unknown quantity <math>x</math> which satisfy the equation. That is to say, to find all the solutions of the equation.
 
Historically, and in current teaching, the study of algebra starts with the solving of equations, such as the quadratic equation above. Then more general questions, such as "does an equation have a solution?", "how many solutions does an equation have?", "what can be said about the nature of the solutions?" are considered. These questions led extending algebra to non-numerical objects, such as [[permutation]]s, [[vector (mathematics)|vectors]], [[matrix (mathematics)|matrices]], and [[polynomial]]s. The structural properties of these non-numerical objects were then formalized into [[algebraic structure]]s such as [[group (mathematics)|groups]], [[ring (mathematics)|rings]], and [[field (mathematics)|fields]].
 
Before the 16th century, mathematics was divided into only two subfields, [[arithmetic]] and [[geometry]]. Even though some methods, which had been developed much earlier, may be considered nowadays as algebra, the emergence of algebra and, soon thereafter, of [[infinitesimal calculus]] as subfields of mathematics only dates from the 16th or 17th century. From the second half of the 19th century on, many new fields of mathematics appeared, most of which made use of both arithmetic and geometry, and almost all of which used algebra.
 
Today, algebra has grown considerably and includes many branches of mathematics, as can be seen in the [[Mathematics Subject Classification]]<ref>{{cite web|url=https://www.ams.org/mathscinet/msc/msc2010.html|title=2010 Mathematics Subject Classification|access-date=2014-10-05|archive-date=2014-06-06|archive-url=https://web.archive.org/web/20140606010248/http://www.ams.org/mathscinet/msc/msc2010.html|url-status=live}}</ref>
where none of the first level areas (two digit entries) are called ''algebra''. Today algebra includes section 08-General algebraic systems, 12-[[Field theory (mathematics)|Field theory]] and [[polynomial]]s, 13-[[Commutative algebra]], 15-[[Linear algebra|Linear]] and [[multilinear algebra]]; [[matrix theory]], 16-[[associative algebra|Associative rings and algebras]], 17-[[Nonassociative ring]]s and [[Non-associative algebra|algebras]], 18-[[Category theory]]; [[homological algebra]], 19-[[K-theory]] and 20-[[Group theory]]. Algebra is also used extensively in 11-[[Number theory]] and 14-[[Algebraic geometry]].


== History ==
== History ==
Early forms of algebra were developed by the [[Babylonian]]s and the [[Ancient Greece|Greek]] geometers such as [[Hero of Alexandria]]. However the word "algebra" is a [[Latin language|Latin]] form of the [[Arabic language|Arabic]] word ''Al-Jabr'' ("casting") and comes from a mathematics book ''Al-Maqala fi Hisab-al Jabr wa-al-Muqabilah'', ("Essay on the Computation of Casting and Equation") written in the [[9th century]] by a [[Iran|Persian]] mathematician, [[Muhammad ibn Mūsā al-Khwārizmī]], who was a [[Islam|Muslim]] born in [[Khwarizm]] in [[Uzbekistan]]. He flourished under Al-Ma'moun in [[Baghdad]], [[Iraq]] through 813-833 AD, and died around 840 AD. The book was brought into [[Europe]] and [[translation|translated]] into Latin in the [[12th century]]. The book was then given the name 'Algebra'. (The ending of the mathematician's name, al-Khwarizmi, was changed into a word easier to say in Latin, and became the English word ''[[algorithm]]'').<ref>{{cite web|title=History of Algebra|url=http://www.ucs.louisiana.edu/~sxw8045/history.htm|publisher=UCS Louisiana|accessdate=11 April 2013|archive-date=9 October 2014|archive-url=https://web.archive.org/web/20141009100628/http://www.ucs.louisiana.edu/~sxw8045/history.htm|url-status=dead}}</ref>
{{Main|History of algebra|Timeline of algebra}}


==Examples==
=== Early history of algebra ===


Here is a simple example of an algebra problem:
[[File:Image-Al-Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala.jpg|thumb|upright=0.8|A page from [[:en:Muhammad ibn Musa al-Khwarizmi|Al-Khwārizmī]]'s ''[[The Compendious Book on Calculation by Completion and Balancing|al-Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala]]'']]


:Sue has 12 candies, and Ann has 24 candies. They decide to share so that they have the same number of candies. How many candies will each have?
The roots of algebra can be traced to the ancient [[Babylonian mathematics|Babylonians]],<ref>{{cite book |last=Struik |first=Dirk J. |year=1987 |title=A Concise History of Mathematics |location=New York |publisher=Dover Publications |isbn=978-0-486-60255-4 |url-access=registration |url=https://archive.org/details/concisehistoryof0000stru_m6j1 }}</ref> who developed an advanced arithmetical system with which they were able to do calculations in an [[algorithm]]ic fashion. The Babylonians developed formulas to calculate solutions for problems typically solved today by using [[linear equation]]s, [[quadratic equation]]s, and [[indeterminate equation|indeterminate linear equations]]. By contrast, most [[Ancient Egyptian mathematics|Egyptians]] of this era, as well as [[Greek mathematics|Greek]] and [[Chinese mathematics]] in the 1st millennium BC, usually solved such equations by geometric methods, such as those described in the ''[[Rhind Mathematical Papyrus]]'', [[Euclid's Elements|Euclid's ''Elements'']], and ''[[The Nine Chapters on the Mathematical Art]]''. The geometric work of the Greeks, typified in the ''Elements'', provided the framework for generalizing formulae beyond the solution of particular problems into more general systems of stating and solving equations, although this would not be realized until [[Mathematics in medieval Islam|mathematics developed in medieval Islam]].<ref>See {{harvnb|Boyer|1991}}.</ref>


These are the steps you can use to solve the problem:
By the time of [[Plato]], Greek mathematics had undergone a drastic change. The Greeks created a [[Greek geometric algebra|geometric algebra]] where terms were represented by sides of geometric objects, usually lines, that had letters associated with them.<ref name=citeboyer>See {{harvnb|Boyer|1991}}, ''Europe in the Middle Ages'', p. 258: "In the arithmetical theorems in Euclid's ''Elements'' VII–IX, numbers had been represented by line segments to which letters had been attached, and the geometric proofs in al-Khwarizmi's ''Algebra'' made use of lettered diagrams; but all coefficients in the equations used in the ''Algebra'' are specific numbers, whether represented by numerals or written out in words. The idea of generality is implied in al-Khwarizmi's exposition, but he had no scheme for expressing algebraically the general propositions that are so readily available in geometry."</ref> [[Diophantus]] (3rd century AD) was an [[Alexandria]]n Greek mathematician and the author of a series of books called ''[[Arithmetica]]''. These texts deal with solving [[algebraic equation]]s,<ref>{{cite book |author-link=Florian Cajori |first=Florian |last=Cajori |year=2010 |url=https://books.google.com/books?id=gZ2Us3F7dSwC&pg=PA34 |title=A History of Elementary Mathematics – With Hints on Methods of Teaching |page=34 |isbn=978-1-4460-2221-4 |access-date=2020-10-15 |archive-date=2021-02-21 |archive-url=https://web.archive.org/web/20210221075950/https://books.google.com/books?id=gZ2Us3F7dSwC&pg=PA34 |url-status=live }}</ref> and have led, in [[number theory]], to the modern notion of [[Diophantine equation]].


# To have the same number of candies, Ann has to give some to Sue. Let ''x'' represent the number of candies Ann gives to Sue.  
Earlier traditions discussed above had a direct influence on the [[History of Iran|Persian]] mathematician [[Muhammad ibn Musa al-Khwarizmi|Muḥammad ibn Mūsā al-Khwārizmī]] (c. 780–850). He later wrote ''[[The Compendious Book on Calculation by Completion and Balancing]]'', which established algebra as a mathematical discipline that is independent of [[geometry]] and [[arithmetic]].<ref>{{Cite book|title=Al Khwarizmi: The Beginnings of Algebra|author=Roshdi Rashed|publisher=Saqi Books|date=November 2009|isbn=978-0-86356-430-7}}</ref>
# Sue's candies, plus ''x'', must be the same as Ann's candies minus ''x''.  This is written as: <tt>12 + x = 24 - x</tt>
# Subtract 12 from both sides of the equation. This gives: <tt>x = 12 - x</tt>.  (What happens on one side of the equals sign must happen on the other side too, for the equation to still be true. So in this case when 12 was subtracted from both sides, there was a middle step of <tt>12 + x - 12 = 24 - x - 12</tt>.  After a person is comfortable with this, the middle step is not written down.)
# Add ''x'' to both sides of the equation. This gives: <tt>2x = 12</tt>
# Divide both sides of the equation by 2. This gives <tt>x = 6</tt>. The answer is six.  If Ann gives Sue 6 candies, they will have the same number of candies.
# To check this, put 6 back into the original equation wherever ''x'' was: <tt>12 + 6 = 24 - 6</tt>
# This gives <tt>18=18</tt>, which is true.  They each now have 18 candies.


With practice, algebra can be used when faced with a problem that is too hard to solve any other way. Problems such as building a freeway, designing a cell phone, or finding the cure for a disease all require algebra.
The [[Hellenistic period|Hellenistic]] mathematicians [[Hero of Alexandria]] and Diophantus<ref>{{cite web|url=http://library.thinkquest.org/25672/diiophan.htm |title=Diophantus, Father of Algebra |access-date=2014-10-05 |url-status=dead |archive-url=https://web.archive.org/web/20130727040815/http://library.thinkquest.org/25672/diiophan.htm |archive-date=2013-07-27}}</ref> as well as [[Indian mathematics|Indian mathematicians]] such as [[Brahmagupta]], continued the traditions of Egypt and Babylon, though Diophantus' ''Arithmetica'' and Brahmagupta's ''[[Brāhmasphuṭasiddhānta]]'' are on a higher level.<ref>{{cite web|url=http://www.algebra.com/algebra/about/history/|title=History of Algebra|access-date=2014-10-05|archive-date=2014-11-11|archive-url=https://web.archive.org/web/20141111040653/http://www.algebra.com/algebra/about/history/|url-status=live}}</ref>{{Better source needed|date=October 2017}} For example, the first complete arithmetic solution written in words instead of symbols,<ref>Mackenzie, Dana. ''The Universe in Zero Words: The Story of Mathematics as Told through Equations'', p. 61 (Princeton University Press, 2012).</ref> including zero and negative solutions, to quadratic equations was described by Brahmagupta in his book ''Brahmasphutasiddhanta,'' published in 628 AD.<ref name="Bradley">Bradley, Michael. ''The Birth of Mathematics: Ancient Times to 1300'', p. 86 (Infobase Publishing 2006).</ref> Later, Persian and [[Arabs|Arab]] mathematicians developed algebraic methods to a much higher degree of sophistication. Although Diophantus and the Babylonians used mostly special ''ad hoc'' methods to solve equations, Al-Khwarizmi's contribution was fundamental. He solved linear and quadratic equations without algebraic symbolism, [[negative numbers]] or [[zero]], thus he had to distinguish several types of equations.<ref name="Meri2004">{{cite book|first=Josef W.|last=Meri|title=Medieval Islamic Civilization|url=https://books.google.com/books?id=H-k9oc9xsuAC&pg=PA31|access-date=2012-11-25|year=2004|publisher=Psychology Press|isbn=978-0-415-96690-0|page=31|archive-date=2013-06-02|archive-url=https://web.archive.org/web/20130602195207/http://books.google.com/books?id=H-k9oc9xsuAC&pg=PA31|url-status=live}}</ref>


== Writing algebra ==
In the context where algebra is identified with the [[theory of equations]], the Greek mathematician Diophantus has traditionally been known as the "father of algebra" and in the context where it is identified with rules for manipulating and solving equations, Persian mathematician al-Khwarizmi is regarded as "the father of algebra".<ref>{{Cite book|last=Corona|first=Brezina|title=Al-Khwarizmi: The Inventor Of Algebra|publisher=Rosen Pub Group|date=February 8, 2006|isbn=978-1404205130|location=New York, United States}}</ref><ref>See {{harvnb|Boyer|1991}}, page 181: "If we think primarily of the matter of notations, Diophantus has good claim to be known as the 'father of algebra', but in terms of motivation and concept, the claim is less appropriate. The Arithmetica is not a systematic exposition of the algebraic operations, or of algebraic functions or of the solution of algebraic equations".</ref><ref>See {{harvnb|Boyer|1991}}, page 230: "The six cases of equations given above exhaust all possibilities for linear and quadratic equations...In this sense, then, al-Khwarizmi is entitled to be known as 'the father of algebra'".</ref><ref>See {{harvnb|Boyer|1991}}, page 228: "Diophantus sometimes is called the father of algebra, but this title more appropriately belongs to al-Khowarizmi".</ref><ref name="Gandz">See {{harvnb|Gandz|1936}}, page 263–277: "In a sense, al-Khwarizmi is more entitled to be called "the father of algebra" than Diophantus because al-Khwarizmi is the first to teach algebra in an elementary form and for its own sake, Diophantus is primarily concerned with the theory of numbers".</ref><ref>{{Cite journal |last=Christianidis |first=Jean |date=August 2007 |title=The way of Diophantus: Some clarifications on Diophantus' method of solution|journal=[[Historia Mathematica]]|volume=34|issue=3|pages=289–305|quote=It is true that if one starts from a conception of algebra that emphasizes the solution of equations, as was generally the case with the Arab mathematicians from al-Khwārizmī onward as well as with the Italian algebraists of the Renaissance, then the work of Diophantus appears indeed very different from the works of those algebraists|doi=10.1016/j.hm.2006.10.003|doi-access=free}}</ref><ref>{{cite journal |first=G. C. |last= Cifoletti |title= La question de l'algèbre: Mathématiques et rhétorique des homes de droit dans la France du 16e siècle |journal= Annales de l'École des Hautes Études en Sciences Sociales, 50 (6)|year= 1995 |pages= 1385–1416 |quote= Le travail des Arabes et de leurs successeurs a privilégié la solution des problèmes.Arithmetica de Diophantine ont privilégié la théorie des equations}}</ref> It is open to debate whether Diophantus or al-Khwarizmi is more entitled to be known, in the general sense, as "the father of algebra". Those who support Diophantus point to the fact that the algebra found in ''Al-Jabr'' is slightly more elementary than the algebra found in ''Arithmetica'' and that ''Arithmetica'' is syncopated while ''Al-Jabr'' is fully rhetorical.<ref>See {{harvnb|Boyer|1991}}, page 228.</ref> Those who support Al-Khwarizmi point to the fact that he introduced the methods of "[[Reduction (mathematics)|reduction]]" and "balancing" (the transposition of subtracted terms to the other side of an equation, that is, the cancellation of [[like terms]] on opposite sides of the equation) which the term ''al-jabr'' originally referred to,<ref name=Boyer-229>See {{harvnb|Boyer|1991}}, ''The Arabic Hegemony'', p. 229: "It is not certain just what the terms ''al-jabr'' and ''muqabalah'' mean, but the usual interpretation is similar to that implied in the translation above. The word ''al-jabr'' presumably meant something like "restoration" or "completion" and seems to refer to the transposition of subtracted terms to the other side of an equation; the word ''muqabalah'' is said to refer to "reduction" or "balancing" – that is, the cancellation of like terms on opposite sides of the equation".</ref> and that he gave an exhaustive explanation of solving quadratic equations,<ref>See {{harvnb|Boyer|1991}}, ''The Arabic Hegemony'', p. 230: "The six cases of equations given above exhaust all possibilities for linear and quadratic equations having positive root. So systematic and exhaustive was al-Khwarizmi's exposition that his readers must have had little difficulty in mastering the solutions".</ref> supported by geometric proofs while treating algebra as an independent discipline in its own right.<ref name="Gandz"/> His algebra was also no longer concerned "with a series of problems to be resolved, but an [[Expository writing|exposition]] which starts with primitive terms in which the combinations must give all possible prototypes for equations, which henceforward explicitly constitute the true object of study". He also studied an equation for its own sake and "in a generic manner, insofar as it does not simply emerge in the course of solving a problem, but is specifically called on to define an infinite class of problems".<ref name=Rashed-Armstrong>{{Cite book |last1= Rashed |first1= R. |last2= Armstrong |first2= Angela |year= 1994 |title= The Development of Arabic Mathematics |publisher= [[Springer Science+Business Media|Springer]] |isbn= 978-0-7923-2565-9 |oclc= 29181926 |pages= 11–12}}</ref>
As in most parts of mathematics, adding ''z'' to ''y'' (or ''y'' plus ''z'') is written as <tt>''y'' + ''z''</tt>.


Subtracting ''z'' from ''y'' (or ''y'' minus ''z'') is written as <tt>''y'' − ''z''</tt>.
Another Persian mathematician [[Omar Khayyam]] is credited with identifying the foundations of [[algebraic geometry]] and found the general geometric solution of the [[cubic equation]]. His book ''Treatise on Demonstrations of Problems of Algebra'' (1070), which laid down the principles of algebra, is part of the body of Persian mathematics that was eventually transmitted to Europe.<ref>{{cite book|title=Mathematical Masterpieces: Further Chronicles by the Explorers |page= 92}}</ref> Yet another Persian mathematician, [[Sharaf al-Dīn al-Tūsī]], found algebraic and numerical solutions to various cases of cubic equations.<ref>{{MacTutor|id=Al-Tusi_Sharaf|title=Sharaf al-Din al-Muzaffar al-Tusi}}</ref> He also developed the concept of a [[Function (mathematics)|function]].<ref>{{Cite journal|last1=Victor J. Katz|first1=Bill Barton|title=Stages in the History of Algebra with Implications for Teaching|journal=Educational Studies in Mathematics|volume=66|issue=2|date=October 2007|doi=10.1007/s10649-006-9023-7|pages=185–201 [192]|last2=Barton|first2=Bill|s2cid=120363574}}</ref> The Indian mathematicians [[Mahavira (mathematician)|Mahavira]] and [[Bhaskara II]], the Persian mathematician [[Al-Karaji]],<ref name="Boyer al-Karkhi ax2n">See {{harvnb|Boyer|1991}}, ''The Arabic Hegemony'', p. 239: "Abu'l Wefa was a capable algebraist as well as a trigonometer.&nbsp;... His successor al-Karkhi evidently used this translation to become an Arabic disciple of Diophantus – but without Diophantine analysis!&nbsp;... In particular, to al-Karkhi is attributed the first numerical solution of equations of the form ax<sup>2n</sup> + bx<sup>n</sup> = c (only equations with positive roots were considered),"</ref> and the Chinese mathematician [[Zhu Shijie]], solved various cases of cubic, [[quartic equation|quartic]], [[quintic equation|quintic]] and higher-order [[polynomial]] equations using numerical methods. In the 13th century, the solution of a cubic equation by [[Fibonacci]] is representative of the beginning of a revival in European algebra. [[Abū al-Ḥasan ibn ʿAlī al-Qalaṣādī]] (1412–1486) took "the first steps toward the introduction of algebraic symbolism". He also computed Σ''n''<sup>2</sup>, Σ''n''<sup>3</sup> and used the method of successive approximation to determine square roots.<ref>{{Cite web|url=http://www-history.mcs.st-andrews.ac.uk/Biographies/Al-Qalasadi.html|title=Al-Qalasadi biography|website=www-history.mcs.st-andrews.ac.uk|access-date=2017-10-17|archive-date=2019-10-26|archive-url=https://web.archive.org/web/20191026001749/http://www-history.mcs.st-andrews.ac.uk/Biographies/Al-Qalasadi.html|url-status=live}}</ref>


Dividing ''y'' by ''z'' (or ''y'' over ''z'': <math>y \over z</math>) is written as <tt>''y'' ÷ ''z''</tt> or <tt>y/z</tt>. <tt>''y''/''z''</tt> is more commonly used.
=== Modern history of algebra ===
[[File:Gerolamo Cardano (colour).jpg|thumb|upright=0.8|Italian mathematician [[Girolamo Cardano]] published the solutions to the [[cubic equation|cubic]] and [[quartic equation]]s in his 1545 book ''[[Ars Magna (Gerolamo Cardano)|Ars magna]]''.]]


In algebra, multiplying ''y'' by ''z'' (or ''y'' times ''z'') can be written in 4 ways: <tt>''y × z''</tt>, <tt>''y * z''</tt>, <tt>''y·z''</tt>, or just <tt>''yz''</tt>. The multiplication symbol "×" is usually not used, because it looks too much like the letter x, which is often used as a variable. Also, when multiplying a larger expression, parentheses can be used: <tt>''y'' (''z+1'')</tt>.
[[François Viète]]'s work on [[new algebra]] at the close of the 16th century was an important step towards modern algebra. In 1637, [[René Descartes]] published ''[[La Géométrie]]'', inventing [[analytic geometry]] and introducing modern algebraic notation. Another key event in the further development of algebra was the general algebraic solution of the cubic and quartic equations, developed in the mid-16th century. The idea of a [[determinant]] was developed by [[Japanese mathematics|Japanese mathematician]] [[Seki Kōwa]] in the 17th century, followed independently by [[Gottfried Leibniz]] ten years later, for the purpose of solving systems of simultaneous linear equations using [[matrix (mathematics)|matrices]]. [[Gabriel Cramer]] also did some work on matrices and determinants in the 18th century. Permutations were studied by [[Joseph-Louis Lagrange]] in his 1770 paper "''Réflexions sur la résolution algébrique des équations''{{-"}} devoted to solutions of algebraic equations, in which he introduced [[Resolvent (Galois theory)|Lagrange resolvents]]. [[Paolo Ruffini]] was the first person to develop the theory of [[permutation group]]s, and like his predecessors, also in the context of solving algebraic equations.


When we multiply a number and a letter in algebra, we write the number in front of the letter: <tt>5 × ''y'' = 5''y''</tt>. When the number is 1, then the 1 is not written because 1 times any number is that number (1 × ''y'' = ''y'') and so it is not needed.
[[Abstract algebra]] was developed in the 19th century, deriving from the interest in solving equations, initially focusing on what is now called [[Galois theory]], and on [[constructible number|constructibility]] issues.<ref>"[http://www.math.hawaii.edu/~lee/algebra/history.html The Origins of Abstract Algebra] {{Webarchive|url=https://web.archive.org/web/20100611091009/http://www.math.hawaii.edu/~lee/algebra/history.html |date=2010-06-11 }}". University of Hawaii Mathematics Department.</ref> [[George Peacock]] was the founder of axiomatic thinking in arithmetic and algebra. [[Augustus De Morgan]] discovered [[relation algebra]] in his ''Syllabus of a Proposed System of Logic''. [[Josiah Willard Gibbs]] developed an algebra of vectors in three-dimensional space, and [[Arthur Cayley]] developed an algebra of matrices (this is a noncommutative algebra).<ref>"[http://www.cambridge.org/catalogue/catalogue.asp?ISBN=978-1108005043 The Collected Mathematical Papers]". Cambridge University Press.</ref>


As a side note, you do not have to use the letters ''x'' or ''y'' in algebra. Variables are just symbols that mean some unknown number or value, so you can use any variable. ''x'' and ''y'' are the most common, though.
== Areas of mathematics with the word algebra in their name ==
[[File:Mathematics lecture at the Helsinki University of Technology.jpg|thumb|[[Linear algebra]] lecture at the [[Aalto University]]]]
Some subareas of algebra have the word algebra in their name; [[linear algebra]] is one example. Others do not: [[group theory]], [[ring theory]], and [[field (mathematics)|field theory]] are examples. In this section, we list some areas of mathematics with the word "algebra" in the name.


== Functions and Graphs ==
* [[Elementary algebra]], the part of algebra that is usually taught in elementary courses of mathematics.
[[File:Linear equation for y=3x+1.png|thumb|Linear equation for y=3x+1]]
* [[Abstract algebra]], in which [[algebraic structure]]s such as [[group (mathematics)|groups]], [[ring (mathematics)|rings]] and [[field (mathematics)|fields]] are [[axiomatization|axiomatically]] defined and investigated.
An important part of algebra is the study of [[Function (mathematics)|functions]], since functions often appear in equations that we are trying to solve. A function is like a machine you can put a number (or numbers) into and get a certain number (or numbers) out. When using functions, [[graph]]s can be powerful tools in helping us to study the solutions to equations.  
* [[Linear algebra]], in which the specific properties of [[linear equation]]s, [[vector space]]s and [[matrix (mathematics)|matrices]] are studied.
* [[Boolean algebra]], a branch of algebra abstracting the computation with the [[truth value]]s ''false'' and ''true''.
* [[Commutative algebra]], the study of [[commutative ring]]s.
* [[Computer algebra]], the implementation of algebraic methods as [[algorithm]]s and [[computer program]]s.
* [[Homological algebra]], the study of algebraic structures that are fundamental to study [[topological space]]s.
* [[Universal algebra]], in which properties common to all algebraic structures are studied.
* [[Algebraic number theory]], in which the properties of numbers are studied from an algebraic point of view.
* [[Algebraic geometry]], a branch of geometry, in its primitive form specifying curves and surfaces as solutions of [[polynomial equation]]s.
* [[Algebraic combinatorics]], in which algebraic methods are used to study combinatorial questions.
* [[Relational algebra]]: a set of [[finitary relation]]s that is [[closure (mathematics)|closed]] under certain operators.


A graph is a picture that shows all the values of the variables that make the equation or inequality true. Usually this is easy to make when there are only one or two variables. The graph is often a line, and if the line does not bend or go straight up-and-down it can be described by the basic formula <tt>y = mx + b</tt>. The variable ''b'' is the [[y-intercept]] of the graph (where the line crosses the vertical axis) and ''m'' is the [[mathematics|slope]] or steepness of the line. This formula applies to the [[Cartesian coordinate system|coordinates]] of a graph, where each point on the line is written <tt>(x, y)</tt>. 
Many mathematical structures are called '''algebras''':


In some math problems like the equation for a line, there can be more than one variable (''x'' and ''y'' in this case).  To find points on the line, one variable is changed. The variable that is changed is called the "independent" variable.  Then the math is done to make a number. The number that is made is called the "dependent" variable. Most of the time the independent variable is written as ''x'' and the dependent variable is written as ''y'', for example, in <tt>y = 3x + 1</tt>.  This is often put on a graph, using an ''x'' axis (going left and right) and a ''y'' axis (going up and down).  It can also be written in function form: <tt>f(x) = 3x + 1</tt>.  So in this example, we could put in 5 for ''x'' and get <tt>y = 16</tt>.  Put in 2 for ''x'' would get <tt>y=7</tt>.  And 0 for ''x'' would get <tt>y=1</tt>.  So there would be a line going thru the points (5,16), (2,7), and (0,1) as seen in the graph to the right.
* [[Algebra over a field]] or more generally [[Algebra (ring theory)|algebra over a ring]].<br />Many classes of algebras over a field or over a ring have a specific name:
** [[Associative algebra]]
** [[Non-associative algebra]]
** [[Lie algebra]]
** [[Composition algebra]]
** [[Hopf algebra]]
** [[C*-algebra]]
** [[Symmetric algebra]]
** [[Exterior algebra]]
** [[Tensor algebra]]
* In [[measure theory]],
** [[Sigma-algebra]]
** [[Algebra over a set]]
* In [[category theory]]
** [[F-algebra]] and [[F-coalgebra]]
** [[T-algebra]]
* In [[logic]],
** [[Relation algebra]], a residuated Boolean algebra expanded with an involution called converse.
** [[Boolean algebra (structure)|Boolean algebra]], a [[complemented lattice|complemented]] [[distributive lattice]].
** [[Heyting algebra]]


If x has a power of 1, it is a straight line.  If it is squared or some other power, it will be curved.  If it uses an inequality (''<'' or ''>''), then usually part of the graph is shaded, either above or below the line.
== Elementary algebra ==
{{main|Elementary algebra}}
[[File:algebraic equation notation.svg|thumb|right|Algebraic expression notation:<br />&nbsp; 1 – power (exponent)<br />&nbsp; 2 – coefficient<br />&nbsp; 3 – term<br />&nbsp; 4 – operator<br />&nbsp; 5 – constant term<br />&nbsp; ''x'' ''y'' ''c'' – variables/constants]]


== Rules ==
Elementary algebra is the most basic form of algebra. It is taught to students who are presumed to have no knowledge of [[mathematics]] beyond the basic principles of [[arithmetic]]. In arithmetic, only [[number]]s and their arithmetical operations (such as +, −, ×, ÷) occur. In algebra, numbers are often represented by symbols called [[variable (mathematics)|variables]] (such as ''a'', ''n'', ''x'', ''y'' or ''z''). This is useful because:
In algebra, there are a few rules that can be used for further understanding of equations. These are called the rules of algebra. While these rules may seem senseless or obvious, it is wise to understand that these properties do not hold throughout all branches of mathematics. Therefore, it will be useful to know how these axiomatic rules are declared, before taking them for granted. Before going on to the rules, reflect on two definitions that will be given.
# Opposite - the opposite of <math>a</math> is <math>-a</math>.
# Reciprocal - the reciprocal of <math>a</math> is <math>\frac{1}{a}</math>.


=== Commutative property of addition ===
* It allows the general formulation of arithmetical laws (such as ''a'' + ''b'' = ''b'' + ''a'' for all ''a'' and ''b''), and thus is the first step to a systematic exploration of the properties of the [[real number|real number system]].
'Commutative' means that a function has the same result if the numbers are swapped around. In other words, the order of the terms in an equation do not matter. When the operator of two terms is an addition, the 'commutative property of addition' is applicable. In algebraic terms, this gives <math>a + b = b + a</math>.
* It allows the reference to "unknown" numbers, the formulation of [[equation]]s and the study of how to solve these. (For instance, "Find a number ''x'' such that 3''x'' + 1 = 10" or going a bit further "Find a number ''x'' such that ''ax'' + ''b'' = ''c''". This step leads to the conclusion that it is not the nature of the specific numbers that allow us to solve it, but that of the operations involved.)
* It allows the formulation of [[function (mathematics)|functional]] relationships. (For instance, "If you sell ''x'' tickets, then your profit will be 3''x'' − 10 dollars, or ''f''(''x'') = 3''x'' − 10, where ''f'' is the function, and ''x'' is the number to which the function is applied".)


Note that this does not apply for subtraction! (i.e. <math>a - b \ne b - a</math>)
=== Polynomials ===
{{main|Polynomial}}
[[File:Polynomialdeg3.svg|The [[graph of a function|graph]] of a polynomial function of degree 3|thumb|upright=0.8]]


=== Commutative property of multiplication ===
A polynomial is an [[expression (mathematics)|expression]] that is the sum of a finite number of non-zero [[Summand|terms]], each term consisting of the product of a constant and a finite number of [[Variable (mathematics)|variables]] raised to whole number powers. For example, ''x''<sup>2</sup> + 2''x'' − 3 is a polynomial in the single variable ''x''. A polynomial expression is an expression that may be rewritten as a polynomial, by using commutativity, associativity and distributivity of addition and multiplication. For example, (''x'' − 1)(''x'' + 3) is a polynomial expression, that, properly speaking, is not a polynomial. A polynomial function is a function that is defined by a polynomial, or, equivalently, by a polynomial expression. The two preceding examples define the same polynomial function.
When the operator of two terms is an multiplication, the 'commutative property of multiplication' is applicable. In algebraic terms, this gives <math>a \cdot b = b \cdot a</math>.


Note that this does not apply for division! (i.e. <math>\frac{a}{b} \ne \frac{b}{a}</math>, when <math>a \neq b </math>)
Two important and related problems in algebra are the [[factorization of polynomials]], that is, expressing a given polynomial as a product of other polynomials that cannot be factored any further, and the computation of [[polynomial greatest common divisor]]s. The example polynomial above can be factored as (''x'' − 1)(''x'' + 3). A related class of problems is finding algebraic expressions for the [[root of a function|roots]] of a polynomial in a single variable.


=== Associative property of addition ===
=== Education ===
'Associative' refers to the grouping of numbers. The associative property of addition implies that, when adding three or more terms, it doesn't matter how these terms are grouped. Algebraically, this gives <math>a + (b + c) = (a + b) + c</math>. Note that this does not hold for subtraction, e.g. <math>1 = 0 - (0 - 1) \neq (0 - 0) - 1 = -1</math> (see the distributive property).
{{see also|Mathematics education}}


=== Associative property of multiplication ===
It has been suggested that elementary algebra should be taught to students as young as eleven years old,<ref>{{Cite web |title=Hull's Algebra |work=[[The New York Times]] |date=July 16, 1904 |url=https://timesmachine.nytimes.com/timesmachine/1904/07/16/101345282.pdf |access-date=2012-09-21 |archive-date=2021-02-21 |archive-url=https://web.archive.org/web/20210221075950/https://timesmachine.nytimes.com/timesmachine/1904/07/16/101345282.pdf |url-status=live }}</ref> though in recent years it is more common for public lessons to begin at the eighth grade level (≈&nbsp;13&nbsp;y.o.&nbsp;±) in the United States.<ref>{{Cite web |last=Quaid |first=Libby |title=Kids misplaced in algebra |publisher=[[Associated Press]] |date=2008-09-22 |url=https://www.usatoday.com/news/nation/2008-09-22-357650952_x.htm |format=Report |access-date=2012-09-23 |archive-date=2011-10-27 |archive-url=https://web.archive.org/web/20111027033958/http://www.usatoday.com/news/nation/2008-09-22-357650952_x.htm |url-status=live }}</ref> However, in some US schools, algebra is started in ninth grade.
The associative property of multiplication implies that, when multiplying three or more terms, it doesn't matter how these terms are grouped. Algebraically, this gives <math>a \cdot (b \cdot c) = (a \cdot b) \cdot c</math>. Note that this does not hold for division, e.g. <math>2 = 1/(1/2) \neq (1/1)/2 = 1/2</math>.


=== Distributive property ===
== Abstract algebra ==
The distributive property states that the multiplication of a number by another term can be distributed. For instance: <math>a \cdot (b + c) = ab + ac</math>. (Do ''not ''confuse this with the associative properties! For instance, <math>a \cdot (b + c) \ne (a \cdot b) + c</math>.)
{{Main|Abstract algebra|Algebraic structure}}


=== Additive identity property ===
Abstract algebra extends the familiar concepts found in elementary algebra and [[arithmetic]] of [[number]]s to more general concepts. Here are the listed fundamental concepts in abstract algebra.
'Identity' refers to the property of a number that it is equal to itself. In other words, there exists an operation of two numbers so that it equals the variable of the sum. The additive identity property states that the sum of any number and 0 is that number: <math>a + 0 = a</math>. This also holds for subtraction: <math>a - 0 = a</math>.


=== Multiplicative identity property ===
[[Set (mathematics)|Sets]]: Rather than just considering the different types of [[number]]s, abstract algebra deals with the more general concept of ''sets'': collections of objects called [[Element (mathematics)|elements]]. All collections of the familiar types of numbers are sets. Other examples of sets include the set of all two-by-two [[Matrix (mathematics)|matrices]], the set of all second-degree [[polynomials]] (''ax''<sup>2</sup> + ''bx'' + ''c''), the set of all two dimensional [[Vector (geometric)|vectors]] of a plane, and the various [[finite groups]] such as the [[cyclic group]]s, which are the groups of integers [[modular arithmetic|modulo]] ''n''. [[Set theory]] is a branch of [[logic]] and not technically a branch of algebra.
The multiplicative identity property states that the product of any number and 1 is that number: <math>a \cdot 1 = a</math>. This also holds for division: <math>\frac{a}{1} = a</math>.


=== Additive inverse property ===
[[Binary operation]]s: The notion of [[addition]] (+) is generalized to the notion of ''binary operation'' (denoted here by ∗). The notion of binary operation is meaningless without the set on which the operation is defined. For two elements ''a'' and ''b'' in a set ''S'', ''a'' ∗ ''b'' is another element in the set; this condition is called [[Closure (mathematics)|closure]]. [[Addition]] (+), [[subtraction]] (−), [[multiplication]] (×), and [[Division (mathematics)|division]] (÷) can be binary operations when defined on different sets, as are addition and multiplication of matrices, vectors, and polynomials.
The additive inverse property is somewhat like the opposite of the additive identity property. When an operation is the sum of a number and its opposite, and it equals 0, that operation is a valid algebraic operation. Algebraically, it states the following: <math>a - a = 0</math>. Additive inverse of 1 is (-1).


=== Multiplicative inverse property ===
[[Identity element]]s: The numbers zero and one are generalized to give the notion of an ''identity element'' for an operation. Zero is the identity element for addition and one is the identity element for multiplication. For a general binary operator ∗ the identity element ''e'' must satisfy ''a'' ∗ ''e'' = ''a'' and ''e'' ∗ ''a'' = ''a'', and is necessarily unique, if it exists. This holds for addition as ''a'' + 0 = ''a'' and 0 + ''a'' = ''a'' and multiplication ''a'' × 1 = ''a'' and 1 × ''a'' = ''a''. Not all sets and operator combinations have an identity element; for example, the set of positive natural numbers (1, 2, 3,&nbsp;...) has no identity element for addition.
The multiplicative inverse property entails that when an operation is the product of a number and its reciprocal, and it equals 1, that operation is a valid algebraic operation. Algebraically, it states the following: <math>\frac{a}{a} = 1</math>. Multiplicative inverse of 2 is 1/2.


==Advanced Algebra==
[[Inverse elements]]: The negative numbers give rise to the concept of ''inverse elements''. For addition, the inverse of ''a'' is written −''a'', and for multiplication the inverse is written ''a''<sup>−1</sup>. A general two-sided inverse element ''a''<sup>−1</sup> satisfies the property that ''a'' ∗ ''a''<sup>−1</sup> = ''e'' and ''a''<sup>−1</sup> ∗ ''a'' = ''e'', where ''e'' is the identity element.
In addition to "[[elementary algebra]]", or basic algebra, there are advanced forms of algebra, taught in colleges and universities, such as [[abstract algebra]], [[linear algebra]], and [[universal algebra]].
This includes how to use a [[matrix (mathematics)|matrix]] to solve many linear equations at once. ''[[Abstract algebra]]'' is the study of things that are found in equations, going beyond numbers to the more abstract with groups of numbers.


Many math problems are about physics and engineering. In many of these physics problems time is a variable. Time uses the letter ''t''. Using the basic ideas in algebra can help reduce a math problem to its simplest form making it easier to solve difficult problems. Energy is ''e'', force is ''f'', mass is ''m'', acceleration is ''a'' and speed of light is sometimes ''c''. This is used in some famous equations, like <tt>f = ma</tt> and <tt>e=mc^2</tt> (although more complex math beyond algebra was needed to come up with that last equation).
[[Associativity]]: Addition of integers has a property called associativity. That is, the grouping of the numbers to be added does not affect the sum. For example: {{nowrap|1=(2 + 3) + 4 = 2 + (3 + 4)}}. In general, this becomes (''a'' ∗ ''b'') ∗ ''c'' = ''a'' ∗ (''b'' ∗ ''c''). This property is shared by most binary operations, but not subtraction or division or [[octonion multiplication]].


==Related pages==
[[Commutative operation|Commutativity]]: Addition and multiplication of real numbers are both commutative. That is, the order of the numbers does not affect the result. For example: 2 + 3 = 3 + 2. In general, this becomes ''a'' ∗ ''b'' = ''b'' ∗ ''a''. This property does not hold for all binary operations. For example, [[matrix multiplication]] and [[Quaternion|quaternion multiplication]] are both non-commutative.
* [[List of mathematics topics]]
* [[Order of Operations]]
* [[Parabola]]
* [[Computer Algebra System]]


==References==
=== Groups ===
{{main|Group (mathematics)}}
{{see also|Group theory|Examples of groups}}
 
Combining the above concepts gives one of the most important structures in mathematics: a [[group (mathematics)|group]]. A group is a combination of a set ''S'' and a single [[binary operation]] ∗, defined in any way you choose, but with the following properties:
 
* An identity element ''e'' exists, such that for every member ''a'' of ''S'', ''e'' ∗ ''a'' and ''a'' ∗ ''e'' are both identical to ''a''.
* Every element has an inverse: for every member ''a'' of ''S'', there exists a member ''a''<sup>−1</sup> such that ''a'' ∗ ''a''<sup>−1</sup> and ''a''<sup>−1</sup> ∗ ''a'' are both identical to the identity element.
* The operation is associative: if ''a'', ''b'' and ''c'' are members of ''S'', then (''a'' ∗ ''b'') ∗ ''c'' is identical to ''a'' ∗ (''b'' ∗ ''c'').
 
If a group is also [[commutativity|commutative]] – that is, for any two members ''a'' and ''b'' of ''S'', ''a'' ∗ ''b'' is identical to ''b'' ∗ ''a'' – then the group is said to be [[Abelian group|abelian]].
 
For example, the set of integers under the operation of addition is a group. In this group, the identity element is 0 and the inverse of any element ''a'' is its negation, −''a''. The associativity requirement is met, because for any integers ''a'', ''b'' and ''c'', (''a'' + ''b'') + ''c'' = ''a'' + (''b'' + ''c'')
 
The non-zero [[rational number]]s form a group under multiplication. Here, the identity element is 1, since 1 × ''a'' = ''a'' × 1 = ''a'' for any rational number ''a''. The inverse of ''a'' is {{sfrac|1|''a''}}, since ''a'' × {{sfrac|1|''a''}} = 1.
 
The integers under the multiplication operation, however, do not form a group. This is because, in general, the multiplicative inverse of an integer is not an integer. For example, 4 is an integer, but its multiplicative inverse is {{sfrac|1|4}}, which is not an integer.
 
The theory of groups is studied in [[group theory]]. A major result of this theory is the [[classification of finite simple groups]], mostly published between about 1955 and 1983, which separates the [[finite set|finite]] [[simple group]]s into roughly 30 basic types.
 
[[Semi-group]]s, [[quasi-group]]s, and [[monoid]]s are [[algebraic structure]]s similar to groups, but with less constraints on the operation. They comprise a set and a closed binary operation but do not necessarily satisfy the other conditions. A [[semi-group]] has an ''associative'' binary operation but might not have an identity element. A [[monoid]] is a semi-group which does have an identity but might not have an inverse for every element. A [[quasi-group]] satisfies a requirement that any element can be turned into any other by either a unique left-multiplication or right-multiplication; however, the binary operation might not be associative.
 
All groups are monoids, and all monoids are semi-groups.
 
{| class="wikitable"
|+Examples
|-
!Set
| colspan=2|[[Natural number]]s '''N'''
| colspan=2|[[Integer]]s '''Z'''
| colspan=4|[[Rational number]]s '''Q'''<br>[[Real number]]s '''R'''<br>[[Complex number]]s '''C'''
| colspan=2|[[Integers modulo n|Integers modulo 3]]<br>'''Z'''/3'''Z''' = {0, 1, 2}
|-
!Operation
| +
| ×
| +
| ×
| +
| −
| ×
| ÷
| +
| ×
|-
!Closed
| Yes
| Yes
| Yes
| Yes
| Yes
| Yes
| Yes
| No
| Yes
| Yes
|-
| Identity
| 0
| 1
| 0
| 1
| 0
| N/A
| 1
| N/A
| 0
| 1
|-
| Inverse
| N/A
| N/A
| −''a''
| N/A
| −''a''
| N/A
| 1/''a''<br>(''a'' ≠ 0)
| N/A
| 0, 2, 1, respectively
| N/A, 1, 2, respectively
|-
| Associative
| Yes
| Yes
| Yes
| Yes
| Yes
| No
| Yes
| No
| Yes
| Yes
|-
| Commutative
| Yes
| Yes
| Yes
| Yes
| Yes
| No
| Yes
| No
| Yes
| Yes
|-
| Structure
| [[monoid]]
| [[monoid]]
| [[abelian group]]
| [[monoid]]
| [[abelian group]]
| [[quasi-group]]
| [[monoid]]
| [[quasi-group]]
| [[abelian group]]
| [[monoid]]
|}
 
=== Rings and fields ===
{{main|Ring (mathematics)|Field (mathematics)}}
{{see also|Ring theory|Glossary of ring theory|Field theory (mathematics)|Glossary of field theory}}
 
Groups just have one binary operation. To fully explain the behaviour of the different types of numbers, structures with two operators need to be studied. The most important of these are [[Ring (mathematics)|rings]] and [[Field (mathematics)|fields]].
 
A [[Ring (mathematics)|ring]] has two binary operations (+) and (×), with × distributive over +. Under the first operator (+) it forms an ''abelian group''. Under the second operator (×) it is associative, but it does not need to have an identity, or inverse, so division is not required. The additive (+) identity element is written as 0 and the additive inverse of ''a'' is written as −''a''.
 
[[Distributivity]] generalises the ''distributive law'' for numbers. For the integers {{nowrap|1=(''a'' + ''b'') × ''c'' = ''a'' × ''c'' + ''b'' × ''c''}} and {{nowrap|1=''c'' × (''a'' + ''b'') = ''c'' × ''a'' + ''c'' × ''b'',}} and × is said to be ''distributive'' over +.
 
The integers are an example of a ring. The integers have additional properties which make it an [[integral domain]].
 
A [[Field (mathematics)|field]] is a ''ring'' with the additional property that all the elements excluding 0 form an ''abelian group'' under ×. The multiplicative (×) identity is written as 1 and the multiplicative inverse of ''a'' is written as ''a''<sup>−1</sup>.
 
The rational numbers, the real numbers and the complex numbers are all examples of fields.
 
== See also ==
{{Portal|Mathematics}}
* [[Outline of algebra]]
* [[Outline of linear algebra]]
* [[Algebra tile]]
 
== References ==
=== Citations ===
{{reflist}}
{{reflist}}


== Other websites ==
=== Works cited ===
* {{cite book |last= Boyer |first= Carl B. |author-link= Carl Benjamin Boyer |title= A History of Mathematics |edition= 2nd |publisher= John Wiley & Sons |year= 1991 |isbn= 978-0-471-54397-8 |url= https://archive.org/details/historyofmathema00boye }}
* {{cite journal |last=Gandz |first=S. |title= The Sources of Al-Khowārizmī's Algebra |journal= [[Osiris (journal)|Osiris]] |volume= 1 |date=January 1936 |pages=263–277 |jstor=301610 |doi=10.1086/368426|s2cid=60770737 }}
* {{cite book |last=Herstein |first=I. N. |year=1964 |title=Topics in Algebra |publisher=Ginn and Company |isbn=0-471-02371-X }}


== Further reading ==
* {{cite book|ref=none |last= Allenby |first= R. B. J. T. |title= Rings, Fields and Groups |isbn= 0-340-54440-6 |year= 1991 }}
* {{cite book|ref=none |last=Asimov|first=Isaac |title=Realm of Algebra|year=1961|publisher=Houghton Mifflin|author-link=Isaac Asimov}}
* {{cite book|ref=none |last= Euler |first=Leonhard |author-link= Leonhard Euler |url= http://web.mat.bham.ac.uk/C.J.Sangwin/euler/ |title= Elements of Algebra |isbn= 978-1-899618-73-6 |archive-url=https://web.archive.org/web/20110413234352/http://web.mat.bham.ac.uk/C.J.Sangwin/euler/ |archive-date=2011-04-13 |date=November 2005 }}
* {{cite book|ref=none |last= Herstein |first=I. N. |title= Topics in Algebra |url= https://archive.org/details/topicsinalgebra00hers |url-access= registration |isbn= 0-471-02371-X |year=1975 }}
* {{cite book|ref=none |last= Hill |first= Donald R. |title= Islamic Science and Engineering |publisher= Edinburgh University Press |year= 1994}}
* {{cite book|ref=none |last= Joseph |first= George Gheverghese |title= The Crest of the Peacock: Non-European Roots of Mathematics |publisher= [[Penguin Books]] |year= 2000}}
* {{cite web |ref=none |last1= O'Connor |first1= John J. |last2= Robertson |first2= Edmund F. |year= 2005 |title= History Topics: Algebra Index |url= http://www-history.mcs.st-andrews.ac.uk/Indexes/Algebra.html |department= [[MacTutor History of Mathematics archive]] |publisher= [[University of St Andrews]] |access-date= 2011-12-10 |archive-url= https://web.archive.org/web/20160303180029/http://www-history.mcs.st-andrews.ac.uk/Indexes/Algebra.html |archive-date= 2016-03-03 |url-status= dead }}
* {{cite book|ref=none |last1=Sardar |first1=Ziauddin |last2=Ravetz |first2=Jerry |last3=Loon |first3=Borin Van |year=1999 |title=Introducing Mathematics |publisher=Totem Books}}
== External links ==
{{Wikiquote}}
{{Wiktionary|algebra}}
{{Wiktionary|algebra}}
{{Wikibooks|Algebra}}
{{Wikibooks|Algebra}}
* [http://www.khanacademy.org/math/algebra Khan Academy: Algebra theory and practice]
{{EB1911 poster|Algebra}}
* [http://algebrarules.com Algebrarules.com: A free place to learn the basics of Algebra]
* [http://www.khanacademy.org/math/algebra Khan Academy: Conceptual videos and worked examples]
* [https://www.khanacademy.org/math/algebra/introduction-to-algebra/overview_hist_alg/v/origins-of-algebra Khan Academy: Origins of Algebra, free online micro lectures]
* [https://www.khanacademy.org/math/algebra/introduction-to-algebra/overview_hist_alg/v/origins-of-algebra Khan Academy: Origins of Algebra, free online micro lectures]
* [http://algebrarules.com Algebrarules.com: An open source resource for learning the fundamentals of Algebra]
* [https://web.archive.org/web/20071004172100/http://www.gresham.ac.uk/event.asp?PageId=45&EventId=620 4000 Years of Algebra], lecture by Robin Wilson, at [[Gresham College]], October 17, 2007 (available for MP3 and MP4 download, as well as a text file).
* {{cite SEP |url-id=algebra |title=Algebra |last=Pratt |first=Vaughan}}
{{Algebra |expanded}}
{{Areas of mathematics |collapsed}}
{{Authority control}}


[[Category:Algebra| ]]
[[Category:Algebra| ]]
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